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Expanding Flow Shop Tasks Based on Recursive Functions

Boris Kupriyanov, Alexander Lazarev, Alexandr Roschin, Frank Werner

Abstract

The paper discusses several extensions of the recursive representation of the flow shop scheduling problem. It is shown that recursive functions make it possible to describe multiple extensions in a single problem. The paper considers altogether six extensions. The examples consider three types of recursive functions: functions associated with the machine, functions that adjust the procession time based on constraints, and functions that control the feasibility of the schedule. The structure of the superpositions of these functions is presented, and also descriptions of several objective functions by recursive functions are presented. Then the general requirements for a recursive function are formulated and its properties are described. Finally, a demonstration of the formulation of new problems is provided using examples of simple flow shop extensions and branch and bound optimization.

Expanding Flow Shop Tasks Based on Recursive Functions

Abstract

The paper discusses several extensions of the recursive representation of the flow shop scheduling problem. It is shown that recursive functions make it possible to describe multiple extensions in a single problem. The paper considers altogether six extensions. The examples consider three types of recursive functions: functions associated with the machine, functions that adjust the procession time based on constraints, and functions that control the feasibility of the schedule. The structure of the superpositions of these functions is presented, and also descriptions of several objective functions by recursive functions are presented. Then the general requirements for a recursive function are formulated and its properties are described. Finally, a demonstration of the formulation of new problems is provided using examples of simple flow shop extensions and branch and bound optimization.
Paper Structure (9 sections, 75 equations, 12 figures, 2 tables)

This paper contains 9 sections, 75 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Description of the flow shop problem class
  3. Representing the PFSP using a recursive function
  4. Examples of recursive functions for extending flow shop problems
  5. Objective functions
  6. Recursive functions
  7. Properties of recursive functions
  8. Branch and bound method
  9. Conclusion

Figures (12)

  • Figure 1: Example of a flow shop problem chain.
  • Figure 2: Example of an expanded graph for a flow shop problem.
  • Figure 3: Illustration of vertex traversal when calculating the recursive function $C(j,k)$ in the flow shop problem.
  • Figure 4: An example of the structure of superpositions of PFSP functions
  • Figure 5: An example of implementing the $r_j$ constraint.
  • ...and 7 more figures